University of Sydney Algebra Seminar

Anna Romanov (University of Sydney)

Friday 23 November, 12-1pm, Place: Carslaw 375

An orbit model for the spectra of nilpotent Gelfand pairs

Let N be a connected and simply connected nilpotent Lie group, and let K be a subgroup of the automorphism group of N. We say that the pair (K,N) is a nilpotent Gelfand pair if the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In 2008, Benson and Racliff established a one-to-one correspondence between the set of bounded K-spherical functions for such a Gelfand pair and a set of K-orbits in the dual of the Lie algebra of N. The sets on either side of this bijection can be given the structure of topological spaces, and Benson and Ratcliff conjectured that this set-theoretic correspondence is actually topological. In this talk, we will describe Benson-Racliff's construction and provide a proof of their conjecture for a certain class of nilpotent Gelfand pairs, establishing a geometric model for the spectrum of such pairs.
This is joint work with H. Friedlander, W. Grodzicki, W. Johnson, G. Ratcliff, B. Strasser, and B. Wessel.